3.9 \(\int \frac{e^{n \cot ^{-1}(a x)}}{(c+a^2 c x^2)^{2/3}} \, dx\)

Optimal. Leaf size=147 \[ -\frac{3 x \left (\frac{1}{a^2 x^2}+1\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (a^2 c x^2+c\right )^{2/3}} \]

[Out]

(-3*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((4 - (3*I)*n)/6)*(1 - I/(a*x))^((-4 + (3*I)*n)/6)*(1 + I/(a
*x))^((2 - (3*I)*n)/6)*x*Hypergeometric2F1[1/3, (4 - (3*I)*n)/6, 4/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(2
/3)

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Rubi [A]  time = 0.196872, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5122, 5126, 132} \[ -\frac{3 x \left (\frac{1}{a^2 x^2}+1\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (a^2 c x^2+c\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(2/3),x]

[Out]

(-3*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((4 - (3*I)*n)/6)*(1 - I/(a*x))^((-4 + (3*I)*n)/6)*(1 + I/(a
*x))^((2 - (3*I)*n)/6)*x*Hypergeometric2F1[1/3, (4 - (3*I)*n)/6, 4/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(2
/3)

Rule 5122

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c + d*x^2)^p/(x^(2*p)*(1
+ 1/(a^2*x^2))^p), Int[u*x^(2*p)*(1 + 1/(a^2*x^2))^p*E^(n*ArcCot[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[d, a^2*c] &&  !IntegerQ[(I*n)/2] &&  !IntegerQ[p]

Rule 5126

Int[E^(ArcCot[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Su
bst[Int[((1 - (I*x)/a)^(p + (I*n)/2)*(1 + (I*x)/a)^(p - (I*n)/2))/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, c, d
, m, n, p}, x] && EqQ[c, a^2*d] &&  !IntegerQ[(I*n)/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !(IntegerQ[2*p] && In
tegerQ[p + (I*n)/2]) &&  !IntegerQ[m]

Rule 132

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x
)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c -
a*d)*(e + f*x)))])/(((b*e - a*f)*(m + 1))*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n), x] /; FreeQ[{a
, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] &&  !IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{e^{n \cot ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{2/3}} \, dx &=\frac{\left (\left (1+\frac{1}{a^2 x^2}\right )^{2/3} x^{4/3}\right ) \int \frac{e^{n \cot ^{-1}(a x)}}{\left (1+\frac{1}{a^2 x^2}\right )^{2/3} x^{4/3}} \, dx}{\left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac{\left (1+\frac{1}{a^2 x^2}\right )^{2/3} \operatorname{Subst}\left (\int \frac{\left (1-\frac{i x}{a}\right )^{-\frac{2}{3}+\frac{i n}{2}} \left (1+\frac{i x}{a}\right )^{-\frac{2}{3}-\frac{i n}{2}}}{x^{2/3}} \, dx,x,\frac{1}{x}\right )}{\left (\frac{1}{x}\right )^{4/3} \left (c+a^2 c x^2\right )^{2/3}}\\ &=-\frac{3 \left (1+\frac{1}{a^2 x^2}\right )^{2/3} \left (\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right )^{\frac{1}{6} (4-3 i n)} \left (1-\frac{i}{a x}\right )^{\frac{1}{6} (-4+3 i n)} \left (1+\frac{i}{a x}\right )^{\frac{1}{6} (2-3 i n)} x \, _2F_1\left (\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left (a+\frac{i}{x}\right ) x}\right )}{\left (c+a^2 c x^2\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.153477, size = 89, normalized size = 0.61 \[ -\frac{3 \sqrt [3]{a^2 c x^2+c} \left (-1+e^{2 i \cot ^{-1}(a x)}\right ) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left (1,\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{4}{3};e^{-2 i \cot ^{-1}(a x)}\right )}{a c (3 n-2 i)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(2/3),x]

[Out]

(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*(c + a^2*c*x^2)^(1/3)*Hypergeometric2F1[1, 2/3 + (
I/2)*n, 4/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c*(-2*I + 3*n))

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Maple [F]  time = 0.291, size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{n{\rm arccot} \left (ax\right )}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccot(a*x))/(a^2*c*x^2+c)^(2/3),x)

[Out]

int(exp(n*arccot(a*x))/(a^2*c*x^2+c)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccot(a*x))/(a^2*c*x^2+c)^(2/3),x, algorithm="maxima")

[Out]

integrate(e^(n*arccot(a*x))/(a^2*c*x^2 + c)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccot(a*x))/(a^2*c*x^2+c)^(2/3),x, algorithm="fricas")

[Out]

integral(e^(n*arccot(a*x))/(a^2*c*x^2 + c)^(2/3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{acot}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acot(a*x))/(a**2*c*x**2+c)**(2/3),x)

[Out]

Integral(exp(n*acot(a*x))/(c*(a**2*x**2 + 1))**(2/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \operatorname{arccot}\left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccot(a*x))/(a^2*c*x^2+c)^(2/3),x, algorithm="giac")

[Out]

integrate(e^(n*arccot(a*x))/(a^2*c*x^2 + c)^(2/3), x)